President Trump arrives at the World Economic Forum last week in Davos, Switzerland. (Laurent Gillieron/Keystone/AP)

While White House Counsel Donald McGahn effectively stopped Trump from trying to fire Mueller in June, Rosenstein is technically the gatekeeper for that decision. With Attorney General Jeff Sessions having recused himself from the Russia probe — a decision that Trump has publicly lamented — Rosenstein has taken oversight of it as the Justice Department's No. 2-ranking official.

That means he has final say on decisions like whether to fire Mueller, with Trump's only recourse being to remove Rosenstein and try to get someone else to do what he wants. If Rosenstein were removed, that “someone else” would either be a replacement deputy attorney general or the No. 3-ranking Justice Department official, Rachel Brand. (Philip Bump has the sequential rundown of who would take oversight of the Russia probe if Trump began firing people who refuse to do his bidding.)

But while firing Mueller appears to be off the table — for now, at least — this person could also hypothetically make more Trump-friendly decisions in other ways.

“He could install someone who would limit Mueller in subtle ways that are defensible,” former federal prosecutor Renato Mariotti said. “Under the special counsel regulations, the attorney general (or acting attorney general, in this case) can ask Mueller for explanations of his actions and overrule them.”

The immediate question would seem to be whether Brand fits the bill. The other big question would be whether Republicans would allow Trump to pick a more sympathetic deputy attorney general.

The Times reported last week that Trump viewed Brand as being preferable to Rosenstein. Whether that's because he knows anything about Brand or simply dislikes Rosenstein — who he's wrongly suggested is a Democrat and has privately said is a threat to his presidency — isn't clear. Rosenstein is a holdover from the Obama administration but was reappointed by Trump and was first appointed in the George W. Bush administration; Brand also served in the Bush administration but not the Obama administration.

“It seems like there is an assumption that Brand can be pushed around on this,” Zeidenberg said. “I have no idea if that is true, but she was never questioned on it during her confirmation and, obviously, made no pledges.”

But Matthew Miller, a former top official in the Obama-era Justice Department, said he doesn't believe Brand would be a pushover.

“She doesn’t have any criminal experience that I’m aware of, and that’s a huge part of the job,” Miller said. “But from everything I know about her, she would follow the law and not be a toady for Trump. She’s fairly well respected.”

Which brings us to Rosenstein's replacement as deputy attorney general. This person would be subject to confirmation by the Republican-controlled Senate, which could push that pick through with a bare majority. But given that Republicans only have an effective 51-49 majority and can lose only two votes one vote, the furor set off by getting rid of Rosenstein would probably make it difficult to replace him with, in Miller's words, a “toady for Trump.”

President Trump has weighed in on special counsel Robert Mueller's investigation into Russian meddling in the 2016 election time and time again. Here's a look at how he can limit the probe, and what Congress is trying to do about it. (Jenny Starrs/The Washington Post)

In the meantime, though, Trump could pick an acting deputy attorney general who could take oversight if he or she didn't recuse himself or herself. Such picks are usually drawn from those who have already won Senate confirmation, especially so they can sign off on surveillance applications (which would limit Trump's options). Brand would be the most logical pick to move into the temporary role as the No. 2 official. But Trump could go a different direction altogether if he wanted.

That said, even appointing an acting deputy attorney general who was viewed as being more sympathetic to Trump would be dicey politically in the aftermath of Rosenstein's ouster. Which means it really depends on how desperate Trump is.

“I guess the million-dollar question is whether Trump wants Rosenstein gone because it would remove an irritant who he sees as responsible for the investigation or because he has some well-considered plan to end it by replacing him with a hack. My guess is the former,” Miller said. “He may have some idea that firing him would hurt the investigation, like he probably had in mind when he fired [then-FBI Director James B.] Comey, but as always, his lack of understanding of the mechanisms of government checks his authoritarian impulses.”

Correction: This post for a time incorrectly described the memo as saying Rosenstein had approved extended surveillance of Page during the 2016 election. This allegedly happened in the spring of 2017, after Rosenstein was confirmed.